Questions about the branch of abstract algebra that deals with groups.

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8
votes
3answers
323 views

Frobenius complement/kernel of an infinite group

Happy Peaceful New Year ! In this question, I recalled that if $H$ is a proper subgroup of a finite group $G$, such that $$({\bf A1})\qquad(g\not\in H)\Longrightarrow(g^{-1}Hg\cap H=(1)),$$ then ...
2
votes
2answers
244 views

Abelian extremely amenable group?

Is there a nontrivial commutative Hausdorff topological group that is extremely amenable? Recall that a topological group is called extremely amenable if any continuous action on a compact ...
3
votes
1answer
112 views

embedding of $O_4^-(q)$ in $U_4(q)$

For $q$ an odd prime power, one can construct $H:=O_4^-(q)<U_4(q)$ by treating the $H$-invariant bilinear form as Hermitean. On the other hand $U_4(q)$ is isomorphic to $O_6^-(q)$; I need to ...
14
votes
2answers
480 views

factorization of the regular representation of the symmetric group

Let $\mathbb{C}[S_n]$ be the regular representation of the symmetric group $S_n$, and let $\mathbb{C}^n$ be the vector representation. Question: Does there exist a representation $V$ (of dimension ...
4
votes
2answers
159 views

Frobenius Groups of Automorphisms

Recently, I am looking different papers on the topic $$\mbox{Frobenius groups of automorphisms of a group.}$$ But I am familiar with Frobenius groups only; not with their (faithful) actions on groups. ...
4
votes
2answers
325 views

A question about generating set of groups and epimorphism

Do there exist non-isomorphic finitely generated groups, $G$ and $H$, along with epimorphisms $\phi:G\rightarrow H$ and $\psi:H\rightarrow G$, such that every generating set of these groups is an ...
8
votes
1answer
165 views

Does every generating set of the first homology group of a Cayley graph give rise to a presentation of its group?

Let $G$ be a group, and fix a symmetric generating set $S$. Let $X$ be the corresponding Cayley graph. Let $R$ be a set of words in $S$, each corresponding to the identity of $G$, such that the set ...
2
votes
2answers
238 views

Behaviour of cohomology groups under extension of scalars

Let $\hat{R}\to R$ be a homomorphism of commutative unital rings and let $\hat{M}$ be an $\hat{R}G$-module for a group $G$. Does the $R$-module isomorphism $$H^n(G,\hat{M}\otimes R)\cong ...
-3
votes
1answer
95 views

Characterization of some finite cyclic groups [closed]

Given a finite cyclic group $G$ with order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $p_i$s are distinct prime numbers $n_i>1$ for all $i$. Let $H$ be any abelian group. Assume that ...
0
votes
1answer
294 views

Which finite groups can be characterized by their automorphism groups?

Given a finite group G, we denote by Aut(G) the group automorphisms of G . Which finite groups G can be characterized by the group Aut(G), i.e. Aut(H)≅Aut(G) implies H≅G?
0
votes
1answer
191 views

Characterizing cyclic group of order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, by Lattice isomorphisms

Given a finite cyclic group $G$ with order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $p_i$s are distinct prime numbers $n_i>1$ for all $i$. Let $H$ be any group.Assume that $L(G)≅L(H)$and ...
2
votes
1answer
75 views

Integer Triples and Reflection tiling $1,2,\ldots,n$

$\forall a,b\in\mathbb Z,\ \exists n\in \mathbb N$ such that the numbers $1,2,\ldots,n$ can be tiled using translates of $\{0,\ a,\ a+b\}$ and $\{0,\ -a,\ -(a+b)\}$ ? In other words for every integer ...
2
votes
0answers
167 views

Manifolds as simultaneous coset spaces

Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...
3
votes
0answers
203 views

Simultaneous coset spaces [closed]

Let $X$ and $Y$ be sets. Under what conditions is there a group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of a point in ...
-1
votes
1answer
167 views

Which finite cyclic groups can be characterized by Lattice isomorphism and isomorphism between their automorphism groups?

Given a finite cyclic group $G$, we denote by $L(G)$ the lattice of its subgroups, and by $\mathop{\rm Aut}(G)$ the automorphism group of $G$. Let $H$ be any group. Assume that $L(G)\cong L(H)$ and ...
0
votes
1answer
218 views

4th Order Floretions: Floret's Equation [closed]

Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information ...
2
votes
2answers
220 views

Maximal size of minimal generating set

Let $G$ be a finite group. Denote by $D(G)$ the maximal size of a minimal generating set, (minimal in the sense of inclusion). I vaguely remember seeing recently something on $D(G)$. Can anyone refer ...
2
votes
0answers
108 views

Which Coxeter groups can be realized as affine reflection groups?

Every affine reflection group has a Coxeter presentation (https://en.wikipedia.org/wiki/Reflection_group#Relation_with_Coxeter_groups). How do you tell which Coxeter presentations arise from affine ...
0
votes
0answers
71 views

When can a 2-cocycle on a subgroup can be extended?

This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective? I am asking this as a new question as I already asked that user but got no ...
5
votes
0answers
59 views

Almost invariance in compact quotients of locally compact groups

While trying to get an analogue of Weiss's monotiling result for amenable residually finite groups in the topological setting, I face the following problem. Let $G$ be a locally compact amenable ...
1
vote
0answers
59 views

Determining finite abelian groups among algebraic theories by counting

Let $T$ be a Lawvere theory (algebraic theory) which contains the theory $T_{\mathrm{grp}}$ of groups as a subtheory (so that $T$ has, in general, more equations that $T_\mathrm{grp}$). Suppose ...
11
votes
1answer
318 views

Can a large transitive permutation group need many generators?

let $G$ be a transitive permutation group acting on $\{1, \ldots, n\}$, and let $d(G)$ be the minimal number of generators of $G$. Is it true, that for $n\rightarrow\infty$ we have ...
0
votes
1answer
104 views

Perfect $Q[G]$-complex

Let $G$ be a finite group and let $M$ be a perfect $\mathbb{Q}[G]$-complex. Suppose that $M\otimes_{\mathbb{Q}[G]}\mathbb{Q}$ is quasi-isomrphic to $0$ can we conclude that $M$ is quasi-isomorphic to ...
3
votes
0answers
202 views

When is the equalizer of group homomorphisms a normal subgroup?

Let $\phi,\psi:G\to H$ be a pair of group homomorphisms. Their equalizer is the subgroup $C\le G$ given by $$ C=\{ g\in G \mid \phi(g)=\psi(g)\} $$ My question is: Under which conditions is the ...
4
votes
1answer
210 views

When the commutativity of the product of group subsets implies the commutativity of the group?

Let $G$ be a finite group. Which are the values of $k$ for which if every two $k$-subsets of $G$ commute then $G$ is commutative? Clearly, this holds for $k=1$ and $k=2$.
12
votes
2answers
232 views

Non-congruence normal subgroups of $SL_2(\mathbb{Z}[1/2])$

Let $G=SL_2(\mathbb{Z}[1/2])$, i.e., the modular group (if you wish) over the ring $\mathbb{Z}[1/2]$ consisting of rationals whose denominators are powers of $2$. Unlike $SL_2(\mathbb{R})$, $G$ is ...
2
votes
0answers
117 views

Acyclicity of covering space

Suppose we have some 2-dimensional non-aspherical finite CW-complex $K$ with $\pi_1(K)=G$. Is there any sufficient condition on $H\leq G$ (and maybe on the group $G$ itself) which allows to conclude ...
6
votes
4answers
616 views

When does $\langle grg^{-1}|g\in G\rangle = \langle gsg^{-1}|g\in G\rangle$ imply $\langle r\rangle=\langle xsx^{-1}\rangle$?

We call a finite group $G$ normal if for all $s,r\in G$, if $\langle gsg^{-1}:g\in G\rangle =\langle grg^{-1}:g\in G\rangle $ then there exists $x\in G$ such that $\langle r\rangle =\langle ...
15
votes
3answers
960 views

Can Calabi-Yau manifolds have nonabelian discrete symmetry groups?

A particle physicist asked me the above question. Let me try to make it more precise. Suppose $M$ is a 3-dimensional Calabi-Yau manifold: that is, a compact Kähler manifold of complex dimension ...
3
votes
1answer
310 views

What do we know about these subgroups of $S_n$?

For each positive integer $n$, write $S_n$ for the symmetric on $n$-letters. Suppose that $m | n$ is a proper divisor of $n$, and write $n = km$. Consider the element $$\displaystyle u(m,n) = ...
0
votes
0answers
82 views

Group which is not MF or AF

Does someone know example of group (countable, discrete) which can not be embedded (monomorphism) into $$ U(\prod M_n/\oplus M_n)$$ unitary group of universal MF-algebra? Or example of group which can ...
-1
votes
1answer
105 views

Notation for a Homomorphism [closed]

Is there a (common) notation which denotes a function, $f$, to be a homomorphism? I have found myself writing, "let $f: X \rightarrow Y$ be a homomorphism" several times. This is fine, but I would ...
2
votes
0answers
68 views

Distal actions on coset spaces

Let $H$ be a group acting by homeomorphisms on a Hausdorff space $X$. Say the action is distal if for all $(x,y) \in X \times X$, if the set $\{(hx,hy) \mid h \in H\}$ accumulates at a diagonal point ...
6
votes
0answers
73 views

Correlation of Class Functions

Let $G$ be a finite group, and let $f_1,f_2$ be two real-valued class functions of $G$. Assume that multiplying elements of $G$ takes $O(1)$-time. Let $s:G\to \mathbb{R}$ be defined by ...
3
votes
1answer
130 views

How to construct a proper action of a group of finite virtual cohomological dimension?

Let $\Gamma$ be the semidirect product of $\mathbb{Z}$ and $\mathbb{Z}/4$, where the action of $\mathbb{Z}/4$ on $\mathbb{Z}$ is defined by $\bar{k} \cdot x = (-1)^k x$. Clearly $\Gamma$ has virtual ...
1
vote
1answer
121 views

Are rotational isometries groups generated by some kind of rotations?

If we consider the index $2$ subgroup of a Weyl group consisting of the isometries with determinant $1$ (the 'special' Weyl group), is it known that it is generated by rotations around some fixed ...
16
votes
2answers
436 views

What categorical property of monoidal categories picks out the ones with duals?

Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...
6
votes
0answers
184 views

Grothendieck - A group as a sheaf over simplicial complexes

In this blog post, Terence Tao gives the following definition of a group. Definition. A group is (identifiable with) a (set-valued) sheaf on the category of simplicial complexes such that the ...
20
votes
2answers
882 views

Why can the general quintic be transformed to $v^5-5\beta v^3+10\beta^2v-\beta^2 = 0$?

The quintic can be transformed to the one-parameter Brioschi quintic, $$u^5-10\alpha u^3+45\alpha^2u-\alpha^2 = 0\tag1$$ This form is well-known for its connection to the symmetries of the ...
7
votes
0answers
250 views

Pedagogical question on Lie groups vs. matrix Lie groups

There are two common approaches taken in introductory texts on Lie groups: studying all Lie groups, or focusing only on matrix Lie groups. The main advantage of the latter approach is that one can ...
-1
votes
1answer
82 views

Group associated to the monoid $({\cal P}(X\times X), \circ)$

Consider the set ${\cal P}(X\times X)$. It can be endowed with a binary operation $\circ$ where $$A\circ B = \{(a,b)\in X\times X:\exists z\in X((a,z)\in A\land (z,b)\in B)\}.$$ Note that ...
1
vote
1answer
50 views

How to claculate the $T$-stable subgroup of second cohomology group

Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1. Let $N=\langle y,w \rangle ...
3
votes
1answer
220 views

Finite groups of order $n$ having exactly $n$ subgroups

Is it known a characterization of finite groups of order $n$ having exactly $n$ subgroups? A supplementary question: are there abelian groups other than the trivial group and $\mathbb{Z}_2$ with ...
5
votes
1answer
106 views

Generalization of a lemma of Livne

Let $H$ be a finite $2$-group. Let $N_{4}(H)$ be the subgroup generated by fourth powers. Let $H_{4}$ be the last term in the short exact sequence $1\rightarrow N_4(H) \rightarrow H \rightarrow H_{4} ...
17
votes
2answers
657 views

How large can the smallest generating set of a group $G$ of order $n$ be?

Let $n$ be a natural number. For every group $G$ of order $n$, denote $d(G)$ : The number of elements of the smallest generating set of $G$ How large is the maximum possible value of $d(G)$ ...
13
votes
3answers
413 views

Explicit construction of an element of ${\rm GL}(2, p)$ of order $p+1$

It is well-known that the order of $GL(2, p)$ is $(p^2-1)(p^2-p) = (p-1)^2(p+1)p.$ It is easy to construct matrices of orders $(p-1)$ and $p$ (diagonal and parabolic, respectively), but the only way ...
0
votes
0answers
81 views

Is there a metric defined on the product space of orthogonal groups?

If one considers just the orthogonal group, then there is a natural metric given by [1]: $$\begin{align} \theta & =\frac{1}{2} \| \log(R_1^{-1}R_2) \| \\ & = \frac{1}{2} \| ...
9
votes
1answer
409 views

Dessins d'enfants and absolute Galois group

I would like to know what is the recent progress about the group homomorphism $$ \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\rightarrow \mathrm{Out}(\hat{F_{2}})$$ ...
5
votes
0answers
70 views

Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$. Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
1
vote
1answer
140 views

The groups with nilpotent Hall $p'$ subgroup

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elements. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on ...